Rigorous Results for the Ground States of the Spin-2 Bose-Hubbard Model

In this work, we prove rigorous theorems for the ground states of the spin-2 Bose-Hubbard model, concerning the magnetic properties, degeneracies and forms of the wave functions. The theorems are highly universal in the sense that they do not depend on the lattice structure (including spatial dimension), particle number and spin-independent interaction.

The spin-2 Bose-Hubbard model is a model for ultracold f = 2 spinor bosonic atoms in optical lattices. We prove that, with c₁ and c₂ being the coefficients of the two spin-dependent interaction terms of the Hamiltonian, the ground state has a maximum total spin when c₁ < 0 and c₂ ≧ 5c₁, while tends to be a singlet if c₁ = 0 and c₂ < 0. When c₁ = c₂ = 0, the model has SU(5) symmetry. Furthermore, the ground-state degeneracies and the forms of wave function are also exactly determined in each coefficient region. Our approach takes the advantage of the symmetry of the Hamiltonian and employs sophisticated mathematical tools including the Perron-Frobenius theorem and the min-max theorem, as well as the representation theory of so(5) Lie algebra.

References
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